Problem: We know that $ 0<\dfrac{3}{\sqrt{10n}} < \dfrac{3}{\sqrt{9n}} = \dfrac{1}{\sqrt{n}}$ for any $n\ge 1$. Considering this fact, what does the direct comparison test say about $\sum\limits_{n=1}^{\infty }\dfrac{3}{\sqrt{10n}}$ ? Choose 1 answer: Choose 1 answer: (Choice A) A The series converges. (Choice B) B The series diverges. (Choice C) C The test is inconclusive.
Solution: $\sum_{n=26}^{\infty }{\frac{1}{\sqrt{n}}}$ is a $p$ -series with $p=\dfrac{1}{2}$, so it diverges. Because our given series is term-by-term less than a divergent series, the direct comparison test does not apply. So the direct comparison test is inconclusive.